Chaos and multiple attractors in a fractal–fractional Shinriki’s oscillator model

Abstract

In this paper, we obtain novel chaotic behaviors using differential and integral operators with power-law, exponential-decay and Mittag-Leffler law for the Shinriki’s oscillator model. We studied the uniqueness and existence of the solutions employing the fixed point postulate. Also, we consider fractal–fractional operators to capture self-similarities for this chaotic attractor. These novel operators predict chaotic behaviors involving the fractal derivative in convolution with power-law, exponential decay law and the Mittag-Leffler function. The generalized model with power-law and Mittag-Leffler kernel was solved numerically via the Adams–Bashforth–Moulton and Adams–Moulton scheme, respectively. For another cases, the numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Numerical simulations for the symmetric and asymmetric cases are obtained to show the applicability and computational efficiency of these methods.